A module of an undirected graph is a set X of nodes such for each node x not in X , either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a directi...

Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension dim(X) of a comparability graph X is the dimension of any transitive orientation of X, and by k-DIM we denote the class of comparability graphs X with dim(X) ≤ k. It is known that the complements of comparability gra...

We sharpen the result that polarity and monopolarity are NP-complete problems by showing that they remain NP-complete if the input graph is restricted to be a 3-colourable comparability graph. We start by presenting a construction reducing 1-3-SAT to monopolarity of 3-colourable comparability graphs. Then we show that polarity is at least as hard as monopolarity for input graphs restricted to a...

A topology τ on the vertices of a comparability graph G is said to be compatible with G if each subgraph H of G is graph-connected if and only if it is a connected subspace of (G,τ). In two previous papers we considered the problem of finding compatible topologies for a given comparability graph and we proved that the nonexistence of infinite paths was a necessary and sufficient condition for t...

We consider two problems pertaining to P4-comparability graphs, namely, the problem of recognizing whether a simple undirected graph is a P4-comparability graph and the problem of producing an acyclic P4-transitive orientation of such a graph. Sequential algorithms for these problems have been presented by Hoàng and Reed and very recently by Raschle and Simon, and by Nikolopoulos and Palios. In...

An undirected graph is a treelike comparability graph if it admits a transitive orientation such that its transitive reduction is a tree. We show that treelike comparability graphs are distance hereditary. Utilizing this property, we give a linear time recognition algorithm. We then characterize permutation graphs that are treelike. Finally, we consider the Partitioning into Bounded Cliques pro...

The complexity of the graph isomorphism problem for trapezoid graphs has been open over a decade. This paper shows that the problem is GI-complete. More precisely, we show that the graph isomorphism problem is GI-complete for comparability graphs of partially ordered sets with interval dimension 2 and height 3. In contrast, the problem is known to be solvable in polynomial time for comparabilit...